DOCSLIB.ORG

Shintani's Zeta Function Is Not a Finite Sum of Euler Products

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Shintani's Zeta Function Is Not a Finite Sum of Euler Products SHINTANI’S ZETA FUNCTION IS NOT A FINITE SUM OF EULER PRODUCTS FRANK THORNE Abstract. We prove that the Shintani zeta function associated to the space of binary cubic forms cannot be written as a finite sum of Euler products. Our proof also extends to several closely related Dirichlet series. This answers a question of Wright [22] in the negative. 1. Introduction In this paper, we will prove that Shintani’s zeta function is not a finite sum of Euler products. We will also prove the same for the Dirichlet series counting cubic fields. Our work is motivated by a beautiful paper of Wright [22]. To illustrate Wright’s work, we recall a classical example, namely the Dirichlet series associated to fundamental discrim- inants. We have s 1 s s ζ(s) s L(s, χ4) (1.1) D− = 1 2− +2 4− + 1 4− , 2 − · ζ(2s) − L(2s, χ4) D>0 X as well as a similar formula for negative discriminants. These formulas may look a bit messy. However, if one combines positive and negative discriminants, one has the beautiful formulas s s s ζ(s) 1 s (1.2) D − = 1 2− +2 4− = Disc(K ) , | | − · ζ(2s) 2 | v |p p [Kv:Qp] 2 X Y X≤ and L(s, χ4) arXiv:1112.1397v3 [math.NT] 9 Nov 2012 s s (1.3) sgn(D) D − = 1 4− . | | − L(2s, χ4) X These are special cases of much more general results. Wright obtains similar formulas for quadratic extensions of any global field k of characteristic not equal to 2, which are in turn n the case n = 2 of formulas for Dirichlet series parameterizing the elements of k×/(k×) . He proves his results by considering twists of the Iwasawa-Tate zeta function (n) n (1.4) ζ (ω, Φ) = ω(t) Φ(t x) d×t A. × × | | ZA /k x k× X∈ We will not explain Wright’s notation here, but suffice it to say that the case n = 1 is the zeta function of Tate’s thesis [17]. This zeta function may be viewed as the zeta function associated to the affine line, viewed as a prehomogeneous vector space of degree 1, on which GL(1) acts by φ(t)x = tnx. Generalizing (1.2) and (1.3), Wright proves that all of these zeta functions can be written as finite linear combinations of Euler products. He further remarks 1 2 FRANK THORNE that “an analogue of [these formulas] for the space of binary cubic forms is currently unknown; although, such a formula would be immensely interesting.” Wright is referring to the Shintani zeta function associated to the space of binary cubic forms. This zeta function was introduced by Shintani [16] and further studied by Datskovsky and Wright [21, 6, 7] and many others. This zeta function is defined as follows: The lattice of integral binary cubic forms is defined by (1.5) V := au3 + bu2v + cuv2 + dv3 : a,b,c,d Z , Z { ∈ } and the discriminant of such a form is given by the usual equation (1.6) Disc(f)= b2c2 4ac3 4b3d 27a2d2 + 18abcd. − − − There is a natural action of GL2(Z) (and also of SL2(Z)) on VZ, given by 1 (1.7) (γ f)(u, v)= f((u, v) γ). · det γ · The Shintani zeta functions are given by the Dirichlet series 1 s (1.8) ξ±(s) := Disc(x) − , Stab(x) | | x SL2(Z) VZ | | ∈Disc(Xx)\>0 ± and Shintani proved that they have analytic continuation to C and satisfy a functional equation. These zeta functions are interesting for a variety of reasons; see, e.g., [18] and [7] for applications to counting cubic extensions of number fields. Also note that Datskovsky and Wright’s work yields an adelic formulation of Shintani’s zeta function, similar to (1.4). Some further motivation for Wright’s question is provided by work of Ibukiyama and Saito [10]. They consider the zeta functions associated to the prehomogeneous vector spaces of n n symmetric matrices, for n > 3 odd. In particular they prove, among many other interesting× results, explicit formulas for these zeta functions as sums of two products of the Riemann zeta function. With these results in mind, one might hope to prove similar such formulas for the zeta functions (1.8). In this note we answer Wright’s question in the negative, and prove that no such formulas exist. Theorem 1.1. Neither of the Shintani zeta functions ξ±(s) defined in (1.8) admits a rep- resentation as a finite sum of Euler products. s k In other words, if we write ξ±(s)= n a(n)n− , we cannot write a(n)= i=1 cibi(n) for real numbers ci and multiplicative functions bi(n). It is interesting to note that these zetaP functions do have representationsP as infinite sums s of Euler products. Of course, there is a tautological such representation, writing n− = s (1+n− ) 1 and thus regarding each term in (1.8) as a difference of Euler products. However, Datskovsky− and Wright [6] proved the much more interesting formula 2 s Rk(2s) (1.9) ξ±(s)= ζ(4s)ζ(6s 1) Disc(k) − , − o(k)| | Rk(4s) Xk SHINTANI’S ZETA FUNCTION IS NOT A FINITE SUM OF EULER PRODUCTS 3 where the sum is over all number fields of degree 1, 2, or 3 (up to isomorphism) of the correct sign; o(k) is equal to 6, 2, 1, or 3 when k is trivial, quadratic, cubic and non-Galois, or cubic 3 and cyclic respectively; and Rk(s) is equal to ζ(s) ,ζ(s)ζk(s),ζk(s),ζk(s) respectively. The proof of Theorem 1.1 is not difficult, and it illustrates an application of recent work of Cohen and Morra ([4]; see also Morra’s thesis [11] for a longer version with more examples). For a fundamental discriminant D, they prove explicit formula for the Dirichlet series 2 s (1.10) ΦD(s) := a(Dn )n− , n X where a(Dn2) is the number of cubic fields of discriminant Dn2. More generally, they prove a formula for the Dirichlet series counting cubic extensions of number fields K/k, such that the normal closure of K contains a fixed quadratic subextension K2/k. The formula is rather complicated, and it involves sums over characters of the 3-part of ray class groups associated to Q(√D, √ 3) (or more generally to K2(√ 3)). However, these formulas take much simpler forms when− we assume k = Q, and when− we further restrict to D for which we can control these class groups. We will apply the following special case of their result: Theorem 1.2 (Cohen-Morra [4]). If D < 0, D 3 (mod 9), and 3 ∤ h(D), then we have ≡ 2 s 1 1 2 2 − (1.11) a(Dn )n = + 1+ s 1+ s . −2 2 3 −3 p n D =1 X ( Yp ) Furthermore, if D > 0, D 3 (mod 9), and 3 ∤ h( D/3), then there exists a cubic field E of discriminant 27D such≡ that − − 2 s 1 1 2 2 − (1.12) a(Dn )n = + 1+ s 1+ s −2 6 3 −3 p n D =1 X ( Yp ) 1 1 ωD(p) + 1 s 1+ s , 3 − 3 −3 p D =1 ( Yp ) where ω (p)=2 if p splits completely in E, and ω (p)= 1 otherwise. D D − Remark. In fact, p splits completely in E if and only if it does so in its Galois closure E(√ 3D), and this version of the condition will be more convenient in our proof. − The equation (1.11) is Corollary 7.9 of [4]. The analogue (1.12) does not appear in [4, 11], so in Section 3 we describe how this formula follows from Cohen and Morra’s work. Moreover, in forthcoming work with Cohen [5] we will generalize (1.12) to any D, positive or negative, and we will also prove an analogous formula for quartic fields having fixed cubic resolvent. We can now summarize our proof of Theorem 1.1, in the easier negative discriminant s case. First observe that n a(n)n− := ξ−(s) ζ(s) cannot be an Euler product: By (1.11), we may find a negative D 3 (mod 9) and− many primes p for which a( D) = 0 and a( Dp2) > 0. P ≡ − − 4 FRANK THORNE To prove that that ξ−(s) ζ(s) is not a sum of k Euler products, for any k > 1, we choose appropriate discriminants D− and primes p such that a( D p2) > 0 if and only if i = j. i i − i j Then a(n) “takes too many parameters to describe,” and we will formalize this using an elementary linear algebra argument. The argument for the positive discriminant case is mildly more complicated, but essen- tially the same. Closing remarks. Although we cheerfully admit that a positive result would be more interesting than the present paper, we submit that our negative result is interesting as well. One motivation for our work comes from our previous paper [20], where we studied the Shintani zeta functions analytically, obtaining (limited) results on the location of the zeroes. These zeta functions essentially fit into the Selberg class, which naturally leads to a host of questions. For example, do most of the zeroes of the Shintani zeta functions lie on the critical line? Work of Bombieri and Hejhal [3] establishes that this is true for certain finite sums of L-functions, conditional on standard hypotheses for these L-functions (including GRH).
Load more

Recommended publications
  • Some Questions About Spaces of Dirichlet Series
    Some questions about spaces of Dirichlet series. Jos´eBonet Instituto Universitario de Matem´aticaPura y Aplicada Universitat Polit`ecnicade Val`encia Eichstaett, June 2017 Jos´eBonet Some questions about spaces of Dirichlet series Dirichlet Series P1 an A Dirichlet Series is an expression of the form n=1 ns with s 2 C. Riemann's zeta function: If s 2 C, 1 1 1 X 1 ζ(s) := 1 + + + ::: = : 2s 3s ns n=1 Euler 1737 considered it for s real: He proved ζ(2) = π2=6, and he relation with the prime numbers: −1 Y 1 ζ(s) = 1 − ; ps p the product extended to all the prime numbers p. Jos´eBonet Some questions about spaces of Dirichlet series Riemann Riemann (1826-1866) and his Memoir \Uber¨ die Anzahl der Primzahlen unter einer gegebenen Gr¨osse", in which he formulated the \Riemann Hypothesis". Jos´eBonet Some questions about spaces of Dirichlet series Riemann The series ζ(s) converges if the real part of s is greater than 1, and ζ(s) has an analytic extension to all the complex plane except s = 1, where it has a simple pole. Obtained the functional equation. s s−1 ζ(s) = 2 π sen(πs=2)Γ(1 − s)ζ(1 − s); s 2 C: Proved that there are no zeros of ζ(s) outside 0 ≤ <(s) ≤ 1, except the trivial zeros s = −2; −4; −6; :::. Jos´eBonet Some questions about spaces of Dirichlet series Riemann's Hypothesis Riemann conjectured that all the non-trivial zeros of ζ(s) are in the line <(s) = 1=2.
    [Show full text]
  • Dirichlet Series
    数理解析研究所講究録 1088 巻 1999 年 83-93 83 Overconvergence Phenomena For Generalized Dirichlet Series Daniele C. Struppa Department of Mathematical Sciences George Mason University Fairfax, VA 22030 $\mathrm{U}.\mathrm{S}$ . A 84 1 Introduction This paper contains the text of the talk I delivered at the symposium “Resurgent Functions and Convolution Equations”, and is an expanded version of a joint paper with T. Kawai, which will appear shortly and which will contain the full proofs of the results. The goal of our work has been to provide a new approach to the classical topic of overconvergence for Dirichlet series, by employing results in the theory of infinite order differential operators with constant coefficients. The possibility of linking infinite order differential operators with gap theorems and related subjects such as overconvergence phenomena was first suggested by Ehrenpreis in [6], but in a form which could not be brought to fruition. In this paper we show how a wide class of overconvergence phenomena can be described in terms of infinite order differential operators, and that we can provide a multi-dimensional analog for such phenomena. Let us begin by stating the problem, as it was first observed by Jentzsch, and subsequently made famous by Ostrowski (see [5]). Consider a power series $f(z)= \sum_{=n0}^{+\infty}anZ^{n}$ (1) whose circle of convergence is $\triangle(0, \rho)=\{z\in C : |z|<\rho\}$ , with $\rho$ given, therefore, by $\rho=\varliminf|a_{n}|^{-1/}n$ . Even though we know that the series given in (1) cannot converge outside of $\Delta=\Delta(0, \rho)$ , it is nevertheless possible that some subsequence of its sequence of partial sums may converge in a region overlapping with $\triangle$ .
    [Show full text]
  • SOLUTIONS of the LINEAR BOLTZMANN EQUATION and SOME DIRICHLET SERIES 1. Introduction Classical Special Functions Are Used In
    SOLUTIONS OF THE LINEAR BOLTZMANN EQUATION AND SOME DIRICHLET SERIES A.V. BOBYLEV(∗) AND I.M. GAMBA(∗∗) Abstract. It is shown that a broad class of generalized Dirichlet series (includ- ing the polylogarithm, related to the Riemann zeta-function) can be presented as a class of solutions of the Fourier transformed spatially homogeneous linear Boltz- mann equation with a special Maxwell-type collision kernel. The result is based on an explicit integral representation of solutions to the Cauchy problem for the Boltz- mann equation. Possible applications to the theory of Dirichlet series are briefly discussed. 1. Introduction Classical special functions are used in different fields of mathematics and its ap- plications. Most of such functions (spherical and cylindrical functions, Hermite and Laguerre polynomials and many others) appear as solutions of some ODEs or PDEs. There is, however, another big class of functions which have no connections with differential equations. The well known examples are the Gamma and Beta functions, the Riemann Zeta-function ζ(z), obtained as an analytic continuation of the Dirichlet series [6] 1 X 1 (1.1) ζ(z) = ; Re z > 1 ; nz n=1 and some similar functions. One more example is a generalization of ζ(z), the so called polylogarithm 1 X xn (1.2) Li (x) = ; jxj < 1 ; z nz n=1 also known as de Longni´ere'sfunction [3]. Hilbert conjectured in the beginning of last century that ζ(z) and other functions of the same type do not satisfy algebraic differential relations [4]. His conjectures were proved in [7, 8]. Other references and new results on differential independence of zeta-type functions can also be found, for example, in the book [6].
    [Show full text]
  • Pell Numbers
    Int. Journal of Math. Analysis, Vol. 7, 2013, no. 38, 1877 - 1884 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2013.35131 On Some Identities and Generating Functions for k- Pell Numbers Paula Catarino 1 Department of Mathematics School of Science and Technology University of Trás-os-Montes e Alto Douro (Vila Real – Portugal) [email protected] Copyright © 2013 Paula Catarino. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We obtain the Binet’s formula for k-Pell numbers and as a consequence we get some properties for k-Pell numbers. Also we give the generating function for k-Pell sequences and another expression for the general term of the sequence, using the ordinary generating function, is provided. Mathematics Subject Classification: 11B37, 05A15, 11B83. Keywords : Pell numbers, k-Pell numbers, Generating functions, Binet’s formula 1. Introduction Some sequences of number have been studied over several years, with emphasis on studies of well-known Fibonacci sequence (and then the Lucas sequence) that is related to the golden ratio. Many papers and research work are dedicated to Fibonacci sequence, such as the work of Hoggatt, in [15] and Vorobiov, in [13], among others and more recently we have, for example, the works of Caldwell et al . in [4], Marques in [7], and Shattuck, in [11]. Also 1 Collaborator of CIDMA and CM-UTAD (Portuguese researches centers). 1878 Paula Catarino relating with Fibonacci sequence, Falc ın et al ., in [14], consider some properties for k-Fibonacci numbers obtained from elementary matrix algebra and its identities including generating function and divisibility properties appears in the paper of Bolat et al., in [3].
    [Show full text]
  • 979 KOJIMA on DOUBLE DIRICHLET SERIES the Object of This Note Is
    1933-1 KOJIMA ON DIRICHLET SERIES 979 KOJIMA ON DOUBLE DIRICHLET SERIES BY C. R. ADAMS The object of this note is to direct attention to the most ex­ tensive and for us one of the most interesting papers by Tet- suzô Kojima, whose death on April 7, 1921, at the early age of thirty-four deprived Japan of an able and promising mathe­ matician. This paper, which has only just now come to our notice* and whose existence may not be generally known, is On the double Dirichlet series, (Science Reports of the Tôhoku Imperial University, (1), (math., phys., chem.), vol. 9 (1920), pp. 351-400). It is devoted mainly to the convergence properties of double Dirichlet series, certain related double integrals, double factorial series, and double binomial coefficient series. Unaware of Kojima's work, Leja and the present writerf have recently considered some of the same questions; it may there­ fore be appropriate to give a brief description of Kojima's mem­ oir and to point out the main features of its relationship to the more recent papers cited. a r A double series ^{ =i a f° which the sequence {Smn} of ai conv partial sums, 5TOn =237/—i ^ ^rges as m, w-*oo and of which each row and column converges has been called regularly convergent by Hardy. If {Smn} is both convergent and bounded, the series may be said to be convergent-bounded. One sees at once that every series regularly convergent is convergent- bounded but that the converse is not true. The notion of regular convergence is basic in Kojima's work; that of convergence- bounded in L, AI, and AIL * An obituary in the Tôhoku Mathematical Journal, vol.
    [Show full text]
  • Dirichlet-Power Series -.:: Natural Sciences Publishing
    J. Ana. Num. Theor. 5, No. 1, 41-48 (2017) 41 Journal of Analysis & Number Theory An International Journal http://dx.doi.org/10.18576/jant/050107 Dirichlet-Power Series Hunduma Legesse Geleta∗ Department of Mathematics, Addis Ababa University, Ethiopia Received: 2 Sep. 2016, Revised: 18 Nov. 2016, Accepted: 23 Nov. 2016 Published online: 1 Jan. 2017 Abstract: In this paper we introduce and investigate a new type of series which we call a ”Dirichlet-power” series. This series includes both Dirichlet series and power series as a special case. It arose during the investigation of series representations of the second order hypergeometric zeta function. The Dirichlet-power series has many properties analogous to Dirichlet series. For a Dirichlet-power series we establish Domain of convergence and show the existence of zero free regions to the right half plane. Keywords: Multiplicative cone b-metric space, fixed point, contractive mapping. 1 Introduction In this paper we introduce and investigate a new type of series which we call a ”Dirichlet-power series”. This The Riemann Zeta function ζ(s) defined by series includes both Dirichlet series and power series as a ∞ special case. As in the case of Dirichlet series it has a 1 kind of abscissa of absolute convergence is attached to it ∑ s n=1 n with a right half plane convergence. On the other hand as a power series it has a kind of radius of convergence with with an integral representation D-shaped region of convergence. This kind of series actually arose during the investigation of series ∞ s−1 ζ 1 x representations of the second order hypergeometric zeta (s)= x dx Γ (s) Z0 e − 1 function.
    [Show full text]
  • Contents 1 Dirichlet Series and the Riemann Zeta Function
    Contents 1 Dirichlet Series and The Riemann Zeta Function 1 2 Primes in Arithmetic Progressions 4 3 Functional equations for ζ(s);L(s; χ) 9 4 Product Formula for ξ(s); ξ(s; χ) 18 5 A Zero-Free Region and the Density of Zeros for ζ(s) 23 6 A Zero-Free Region and the Density of Zeros for L(s; χ) 25 7 Explicit Formula Relating the Primes to Zeros of ζ(s) 25 8 Chebyshev Estimates and the Prime Number Theorem 28 9 Prime Number Theorem in Arithmetic Progressions 33 [Sections 1,2, and 3 are OK - the rest need work/reorganization] 1 Dirichlet Series and The Riemann Zeta Function Throughout, s = σ + it is a complex variable (following Riemann). Definition. The Riemann zeta function, ζ(s), is defined by X 1 Y ζ(s) = = (1 − p−s)−1 ns n p for σ > 1. Lemma (Summation by Parts). We have q q−1 X X anbn = An(bn − bn+1) + Aqbq − Ap−1bp n=p n=p P P where An = k≤n ak. In particular, if n≥1 anbn converges and Anbn ! 0 as n ! 1 then X X anbn = An(bn − bn+1): n≥1 n≥1 P Another formulation: if a(n) is a funciton on the integers, A(x) = n≤x a(n), and f is C1 on [y; x] then X Z x a(n)f(n) = A(x)f(x) − A(y)f(y) − A(t)f 0(t)dt y<n≤x y Proof. q q q q X X X X anbn = (An − An−1)bn = Anbn − An−1bn n=p n=p n=p n=p q q−1 q−1 X X X = Anbn − Anbn+1 = An(bn − bn+1) + Aqbq − Ap−1bp: n=p n=p−1 n=p For the second forumulation, assume y is not an integer and let N = dye;M = bxc.
    [Show full text]
  • Some Periodicities in the Continued Fraction Expansions of Fibonacci and Lucas Dirichlet Series
    SOME PERIODICITIES IN THE CONTINUED FRACTION EXPANSIONS OF FIBONACCI AND LUCAS DIRICHLET SERIES CHRIS K. CALDWELL AND TAKAO KOMATSU P1 ¡s Abstract. In this paper we consider the Fibonacci Zeta functions ³F (s) = n=1 Fn and P1 ¡s the Lucas Zeta functions ³L(s) = n=0 Ln . The sequences fAº gº¸0 and fBº gº¸0, which Pn ¡s are derived from º=1 Fº = An=Bn, satisfy certain recurrence formulas. We examine some properties of the periodicities of An and Bn. For example, let m and k be positive m integers. If n ¸ mk, then Bn ´¥ 0¦ (mod Fk¥) (with¦ a similar result holding for An). The n P1 n power of 2 which divides Bn is 6 + i=0 3¢2i . 1. Introduction Consider the so-called Fibonacci and Lucas Zeta functions: X1 1 X1 1 ³ (s) = ; ³ (s) = : F F s L Ls n=1 n n=0 n In [13] the analytic continuation of these series is discussed. In [4] it is shown that the numbers ³F (2); ³F (4); ³F (6) ( respectively, ³L(2); ³L(4); ³L(6)) are algebraically independent, and that each of ³F (2s) ( respectively, ³L(2s)) (s = 4; 5; 6;::: ) may be written as a rational (respectively, algebraic) function of these three numbers over Q, e.g. ³ 15 1 6 5 4 3 ³F (8) ¡ ³F (4) = 256u ¡ 3456u + 2880u + 1792u v 14 378(4u + 5)2 ´ ¡ 11100u3 + 20160u2v ¡ 10125u2 + 7560uv + 3136v2 ¡ 1050v ; where u = ³F (2) and v = ³F (6). Similar results are obtained in [4] for the alternating sums à ! X1 (¡1)n+1 X1 (¡1)n+1 ³¤ (2s) := respectively, ³¤ (2s) := (s = 1; 2; 3;::: ): F F 2s L L2s n=1 n n=1 n From the main theorem in [5] it follows that for any positive distinct integers s1; s2; s3 the numbers ³F (2s1), ³F (2s2), and ³F (2s3) are algebraically independent if and only if at least one of s1; s2; s3 is even.
    [Show full text]
  • Convergence of Dirichlet Series and Euler Products Peter Zvengrowski
    Convergence of Dirichlet Series and Euler Products Peter Zvengrowski 1 Introduction These notes are based on lectures given by the author in 2014 at the Uni- versity of Calgary and in 2015 at the University of N. Carolina Greensboro. The general theme is convergence, in Section 2 this is studied for Dirichlet series and in Sections 3-4 for Euler products. Section 5 gives some examples and concludes with a few questions. 2 Dirichlet Series By a Dirichlet series we mean an infinite series ∞ ∞ a f(s)= n = a n−s, a ∈ C. ns n n n=1 n=1 X X As usual we write s = σ+it for the complex variable s. A very familiar exam- ple is the case an = 1, f is then the Riemann zeta function ζ. For t = 0, i.e. for s = σ ∈ R , it is proved in elementary calculus that ζ(σ) diverges for σ =1 and is absolutely convergent for σ > 1. This is called the “p-test” (where p = σ) but should really be called the “ζ-test.” Another familiar example (again when s = σ ∈ R) is the alternating zeta series η(s)= (−1)n−1n−s, known as the Euler-Dedekind function. It is proved in elementary calculus that this series converges for σ > 0, where the convergence isP conditional for 0 < σ ≤ 1 and absolute for 1 < σ. In this section we shall prove that very similar results hold, with appropriate hypotheses on the coefficients an, for s ∈ C, i.e. dropping the condition t = 0. From elementary complex analysis, for any x ∈ R+, one has |xs| = xσ.
    [Show full text]
  • The Zeta-Regularized Product of Odious Numbers
    The zeta-regularized product of odious numbers J.-P. Allouche CNRS, IMJ-PRG Sorbonne Universit´e 4 Place Jussieu F-75252 Paris Cedex 05 (France) Abstract What is the product of all odious integers, i.e., of all integers whose binary expansion contains an odd number of 1’s? Or more precisely, how to define a product of these integers which is not infinite, but still has a “reasonable” definition? We will answer this question by proving that this product is equal to π1/4 2ϕe−γ, where γ and ϕ are respectively the Euler-Mascheroni and the Flajolet-Martin constants. p — Dedicated to Joseph Kung 1 Introduction Extending or generalizing “simple” notions is a basic activity in mathematics. This involves trying to give some sense to an a priori meaningless formula, like √ 1, 1/0, n, etc. Among these − n≥1 attempts is the question of “assigning a reasonable value” to an infinite product of increasing P positive real numbers. This question arises for example when trying to define “determinants” for operators on infinite-dimensional vector spaces. One possible approach is to define “zeta- regularization” (see the definition in Section 2 below). The literature on the subject is vast, going from theoretical aspects to explicit computations in mathematics but also in physics (see, e.g., [10, 23]): we will —of course— not give a complete view of the existing references, but rather restrict to a few ones to allude to general contexts where these infinite products take place. Our purpose is modest: to give the value of an infinite arithmetic product (namely the product of all arXiv:1906.10532v2 [math.NT] 17 Sep 2019 odious integers, i.e., of all those integers whose binary expansion contains an odd number of 1’s, see, e.g., [19]) 1 2 4 7 8 11 13 14 16 19 21 22 25 26 28 31 32 .
    [Show full text]
  • Multiple Dirichlet Series and Automorphic Forms
    Proceedings of Symposia in Pure Mathematics Multiple Dirichlet Series and Automorphic Forms Gautam Chinta, Solomon Friedberg, and Jeffrey Hoffstein Abstract. This article gives an introduction to the multiple Dirichlet series arising from sums of twisted automorphic L-functions. We begin by explaining how such series arise from Rankin-Selberg constructions. Then more recent work, using Hartogs’ continuation principle as extended by Bochner in place of such constructions, is described. Applications to the nonvanishing of L- functions and to other problems are also discussed, and a multiple Dirichlet series over a function field is computed in detail. 1. Motivation Of the major open problems in modern mathematics, the Riemann hypothesis, which states that the nontrivial zeroes of the Riemann zeta function ζ(s) lie on the 1 line <(s) = 2 , is one of the deepest and most profoundly important. A consequence of the Riemann Hypothesis which has far reaching applications is the Lindel¨of Hypothesis. This states that for any > 0 there exists a constant C() such that for all t, |ζ(1/2 + it)| < C()|t|. The Lindel¨ofHypothesis remains as unreachable today as it was 100 years ago, but there has been a great deal of progress in obtaining approximations of it. These are results of the form |ζ(1/2 + it)| < C()|t|κ+, where κ > 0 is some fixed real number. For example, Riemann’s functional equation for the zeta function, together with Stirling’s approximation for the gamma function and the Phragmen-Lindel¨of principle, are sufficient to obtain what is known as the convexity bound for the zeta 1 1 1 4 + function, namely κ = 4 , or: ζ 2 + it < C()|t| .
    [Show full text]
  • Dirichlet's Theorem on Arithmetic Progressions
    Dirichlet’s Theorem on Arithmetic Progressions Anthony V´arilly Harvard University, Cambridge, MA 02138 1 Introduction Dirichlet’s theorem on arithmetic progressions is a gem of number theory. A great part of its beauty lies in the simplicity of its statement. Theorem 1.1 (Dirichlet). Let a, m ∈ Z, with (a, m) = 1. Then there are infinitely many prime numbers in the sequence of integers a, a + m, a + 2m, . , a + km, . for k ∈ N. A sixth grader knows enough mathematics to understand this particular formulation of the theorem. However, many deep ideas of algebra and analysis are required to prove it. In order to motivate some of the ideas we will introduce, we will sketch how to show there are infinitely many primes of the form 4k + 1, the special case a = 1, m = 4 of Theorem 1.1. We shall follow Knapp’s exposition in our sketch [2]. Define the (real valued) Riemann zeta function as ∞ X 1 ζ(s) = , s > 1. (1) ns n=1 Throughout this paper, p shall denote a prime number, unless otherwise indicated. It is possible to write the zeta function as the infinite product Y 1 ζ(s) = . (2) 1 − p−s p To see why this is true, notice that for finite N, Y 1 X 1 = 1 − p−s ns p≤N n∈S where S is the set of natural numbers whose prime factors do not exceed N. Letting N → ∞ we obtain the result. With this product formula for ζ(s), it is possible to show (and we will do so in the proof of Theorem 1.1) that X 1 log ζ(s) = + g(s) (3) ps p 1 where g(s) is bounded as s → 1.
    [Show full text]